Counting Time Overlaps

There are instructions at the top of the page explaining how to attach your sample workbook.

assuming a version prior to O365 and your sample file, you might try something like:

Formula:

=MAX(COUNTIFS($B$2:$B$7,"<="&TIME(0,ROW($1:$1440),0),$C$2:$C$7,">="&TIME(0,ROW($1:$1440),0)))

confirmed with CTRL + SHIFT + ENTER

this isn't particularly efficient though - as it's performing 1440 COUNTIFS, one for each minute of the day -- you could streamline a little, by isolating MIN & MAX, e.g.:

Formula:

=MAX(COUNTIFS($B$2:$B$7,"<="&TIME(0,ROW(INDEX($A:$A,MIN($B$2:$B$7)*1440):INDEX($A:$A,MAX($C$2:$C$7)*1440)),0),$C$2:$C$7,">="&TIME(0,ROW(INDEX($A:$A,MIN($B$2:$B$7)*1440):INDEX($A:$A,MAX($C$2:$C$7)*1440)),0)))

confirmed with CTRL + SHIFT + ENTER